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Worked solution

Distinguish term limits from series limits

Use limit notation to rewrite: (a) as n→∞n\to\infty, un=(12)n−1→0u_n=(\frac12)^{n-1}\to0; (b) Sn→2S_n\to2 as n→∞n\to\infty, where Sn=1+12+14+⋯+(12)n−1S_n=1+\frac12+\frac14+\cdots+(\frac12)^{n-1}.

CJC · Sequences and Series · Limit-notation question

(a) Limit of the term sequence

Answer

lim⁡n→∞un=lim⁡n→∞(12)n−1=0\displaystyle\lim_{n\to\infty}u_n=\lim_{n\to\infty}(\tfrac12)^{n-1}=0.

Put the limiting instruction before the expression

lim⁡n→∞un=lim⁡n→∞(12)n−1=0\lim_{n\to\infty}u_n=\lim_{n\to\infty}\left(\frac12\right)^{n-1}=0

This answer is already printed in the Student copy. It concerns one term, not the accumulated sum.

How did this part go?

(b) Limit of the partial sums

Answer

lim⁡n→∞Sn=2\displaystyle\lim_{n\to\infty}S_n=2.

Name the partial-sum limit

lim⁡n→∞Sn=2\lim_{n\to\infty}S_n=2

The Teacher completes the blank Student answer and labels this the limit of a series.

How did this part go?

Common mistakes

Terms tending to zero alone do not prove convergence of a series. Here the finite GP formula gives S_n=2[1−(1/2)^n], which tends to 2.